In the 1970s, Erdős asked how many edges are needed in a graph on $n$ vertices, to ensure the existence of a cycle of length exactly $n-k$. In this paper, we consider the spectral analog of Erdős' problem. Indeed, the problem of determining tight spectral radius conditions for cycles of length $ell$ in a graph of order $n$ for each $ell in[3,n]$ seems very difficult. We determine tight spectral radius conditions for $C_{ell}$ where $ell$ belongs to an interval of the form $[n-Theta(sqrt{n}),n]$. As a main tool, we prove a stability result of a theorem due to Woodall, which states that for a graph $G$ of order $ngeq 2k+3$ where $kgeq 0$ is an integer, if $e(G)>binom{n-k-1}{2}+binom{k+2}{2}$ then $G$ contains a $C_{ell}$ for each $ellin [3,n-k]$. We prove a tight spectral condition for the circumference of a $2$-connected graph with a given minimum degree, of which the main tool is a stability version of a 1976 conjecture of Woodall on circumference of a $2$-connected graph with a given minimum degree proved by Ma and the second author. We also give a brief survey on this area and point out where we are and our predicament.
{"title":"Stability of Woodall's Theorem and Spectral Conditions for Large Cycles","authors":"Binlong Li, Bo Ning","doi":"10.37236/11641","DOIUrl":"https://doi.org/10.37236/11641","url":null,"abstract":"In the 1970s, Erdős asked how many edges are needed in a graph on $n$ vertices, to ensure the existence of a cycle of length exactly $n-k$. In this paper, we consider the spectral analog of Erdős' problem. Indeed, the problem of determining tight spectral radius conditions for cycles of length $ell$ in a graph of order $n$ for each $ell in[3,n]$ seems very difficult. We determine tight spectral radius conditions for $C_{ell}$ where $ell$ belongs to an interval of the form $[n-Theta(sqrt{n}),n]$. As a main tool, we prove a stability result of a theorem due to Woodall, which states that for a graph $G$ of order $ngeq 2k+3$ where $kgeq 0$ is an integer, if $e(G)>binom{n-k-1}{2}+binom{k+2}{2}$ then $G$ contains a $C_{ell}$ for each $ellin [3,n-k]$. We prove a tight spectral condition for the circumference of a $2$-connected graph with a given minimum degree, of which the main tool is a stability version of a 1976 conjecture of Woodall on circumference of a $2$-connected graph with a given minimum degree proved by Ma and the second author. We also give a brief survey on this area and point out where we are and our predicament.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"25 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78137996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We make progress toward a characterization of the graphs $H$ such that every connected $H$-free graph has a longest path transversal of size $1$. In particular, we show that the graphs $H$ on at most $4$ vertices satisfying this property are exactly the linear forests. We also show that if the order of a connected graph $G$ is large relative to its connectivity $kappa(G)$, and its independence number $alpha(G)$ satisfies $alpha(G) le kappa(G) + 2$, then each vertex of maximum degree forms a longest path transversal of size $1$.
我们在图$H$的表征方面取得了进展，使得每个连接的$H$自由图都有一个长度为$1$的最长路径截线。特别地，我们证明了在最多$4$个满足此性质的顶点上的图$H$正是线性森林。我们还证明了如果连通图的阶数$G$相对于其连通性$kappa(G)$较大，且其独立数$alpha(G)$满足$alpha(G) le kappa(G) + 2$，则每个最大度顶点形成一个长度为$1$的最长路径截线。
{"title":"Non-Empty Intersection of Longest Paths in $H$-Free Graphs","authors":"James A. Long Jr., Kevin G. Milans, Andrea Munaro","doi":"10.37236/11277","DOIUrl":"https://doi.org/10.37236/11277","url":null,"abstract":"We make progress toward a characterization of the graphs $H$ such that every connected $H$-free graph has a longest path transversal of size $1$. In particular, we show that the graphs $H$ on at most $4$ vertices satisfying this property are exactly the linear forests. We also show that if the order of a connected graph $G$ is large relative to its connectivity $kappa(G)$, and its independence number $alpha(G)$ satisfies $alpha(G) le kappa(G) + 2$, then each vertex of maximum degree forms a longest path transversal of size $1$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136166514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recent works of Andrews–Newman, and Hopkins–Sellers unveil an interesting relation between two partition statistics, the crank and the mex. They state that, for a positive integer $n$, there are as many partitions of $n$ with non-negative crank as partitions of n with odd mex. In this paper, we give a bijective proof of a generalization of this identity provided by Hopkins, Sellers and Stanton. Our method uses an alternative definition of the Durfee decomposition, whose combinatorial link to the crank was recently studied by Hopkins, Sellers, and Yee.
{"title":"A Bijective Proof of a Generalization of the Non-Negative Crank-Odd Mex Identity","authors":"Isaac Konan","doi":"10.37236/11472","DOIUrl":"https://doi.org/10.37236/11472","url":null,"abstract":"Recent works of Andrews–Newman, and Hopkins–Sellers unveil an interesting relation between two partition statistics, the crank and the mex. They state that, for a positive integer $n$, there are as many partitions of $n$ with non-negative crank as partitions of n with odd mex. In this paper, we give a bijective proof of a generalization of this identity provided by Hopkins, Sellers and Stanton. Our method uses an alternative definition of the Durfee decomposition, whose combinatorial link to the crank was recently studied by Hopkins, Sellers, and Yee.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"14 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85295117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that if $A$ is a subset of a group of prime order $p$ such that $|2A|<2.7652|A|$ and $|A|<1.25cdot10^{-6}p$, then $A$ is contained in an arithmetic progression with at most $|2A|-|A|+1$ terms, and $2A$ contains an arithmetic progression with the same difference and at least $2|A|-1$ terms. This improves a number of previously known results towards the conjectured value $3|A|-4$ for which such an statement should hold..
我们证明了如果$A$是一组素数阶$p$的子集，使得$|2A|<2.7652|A|$和$|A|<1.25cdot10^{-6}p$，则$A$包含在一个等差数列中，其项最多为$|2A|- A|+1$，且$2A$包含一个等差数列，且项至少为$2|A|-1$。这改进了许多先前已知的结果，使其趋向于假设值$3| a |-4$，这样的语句应该成立。
{"title":"Towards $3n-4$ in groups of prime order","authors":"V. Lev, O. Serra","doi":"10.37236/11976","DOIUrl":"https://doi.org/10.37236/11976","url":null,"abstract":"We show that if $A$ is a subset of a group of prime order $p$ such that $|2A|<2.7652|A|$ and $|A|<1.25cdot10^{-6}p$, then $A$ is contained in an arithmetic progression with at most $|2A|-|A|+1$ terms, and $2A$ contains an arithmetic progression with the same difference and at least $2|A|-1$ terms. This improves a number of previously known results towards the conjectured value $3|A|-4$ for which such an statement should hold..","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"35 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89478038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The minimum positive $ell$-degree $delta^+_{ell}(G)$ of a non-empty $k$-graph $G$ is the maximum $m$ such that every $ell$-subset of $V(G)$ is contained in either none or at least $m$ edges of~$G$; let $delta^+_{ell}(G):=0$ if $G$ has no edges. For a family $mathcal F$ of $k$-graphs, let $mathrm{co^{+}ex}_ell(n,mathcal F)$ be the maximum of $delta^+_{ell}(G)$ over all $mathcal F$-free $k$-graphs $G$ on $n$ vertices. We prove that the ratio $mathrm{co^{+}ex}_ell(n,mathcal F)/{n-ellchoose k-ell}$ tends to limit as $ntoinfty$, answering a question of Halfpap, Lemons and Palmer. Also, we show that the limit can be obtained as the value of a natural optimisation problem for $k$-hypergraphons; in fact, we give an alternative description of the set of possible accumulation points of almost extremal $k$-graphs.
{"title":"On the Limit of the Positive $ell$-Degree Turán Problem","authors":"O. Pikhurko","doi":"10.37236/11912","DOIUrl":"https://doi.org/10.37236/11912","url":null,"abstract":"The minimum positive $ell$-degree $delta^+_{ell}(G)$ of a non-empty $k$-graph $G$ is the maximum $m$ such that every $ell$-subset of $V(G)$ is contained in either none or at least $m$ edges of~$G$; let $delta^+_{ell}(G):=0$ if $G$ has no edges. For a family $mathcal F$ of $k$-graphs, let $mathrm{co^{+}ex}_ell(n,mathcal F)$ be the maximum of $delta^+_{ell}(G)$ over all $mathcal F$-free $k$-graphs $G$ on $n$ vertices. We prove that the ratio $mathrm{co^{+}ex}_ell(n,mathcal F)/{n-ellchoose k-ell}$ tends to limit as $ntoinfty$, answering a question of Halfpap, Lemons and Palmer. Also, we show that the limit can be obtained as the value of a natural optimisation problem for $k$-hypergraphons; in fact, we give an alternative description of the set of possible accumulation points of almost extremal $k$-graphs.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"15 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85148827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-14DOI: 10.48550/arXiv.2302.07110
James A. Long, K. Milans, Andrea Munaro
We make progress toward a characterization of the graphs $H$ such that every connected $H$-free graph has a longest path transversal of size $1$. In particular, we show that the graphs $H$ on at most $4$ vertices satisfying this property are exactly the linear forests. We also show that if the order of a connected graph $G$ is large relative to its connectivity $kappa(G)$, and its independence number $alpha(G)$ satisfies $alpha(G) le kappa(G) + 2$, then each vertex of maximum degree forms a longest path transversal of size $1$.
我们在图$H$的表征方面取得了进展，使得每个连接的$H$自由图都有一个长度为$1$的最长路径截线。特别地，我们证明了在最多$4$个满足此性质的顶点上的图$H$正是线性森林。我们还证明了如果连通图的阶数$G$相对于其连通性$kappa(G)$较大，且其独立数$alpha(G)$满足$alpha(G) le kappa(G) + 2$，则每个最大度顶点形成一个长度为$1$的最长路径截线。
{"title":"Non-empty intersection of longest paths in H-free graphs","authors":"James A. Long, K. Milans, Andrea Munaro","doi":"10.48550/arXiv.2302.07110","DOIUrl":"https://doi.org/10.48550/arXiv.2302.07110","url":null,"abstract":"We make progress toward a characterization of the graphs $H$ such that every connected $H$-free graph has a longest path transversal of size $1$. In particular, we show that the graphs $H$ on at most $4$ vertices satisfying this property are exactly the linear forests. We also show that if the order of a connected graph $G$ is large relative to its connectivity $kappa(G)$, and its independence number $alpha(G)$ satisfies $alpha(G) le kappa(G) + 2$, then each vertex of maximum degree forms a longest path transversal of size $1$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"213 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73231350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We obtain a characterization of lattice cubes as the only sets that reach equality in several discrete isoperimetric-type inequalities associated with the $L_{infty}$ norm, including well-known results by Radcliffe and Veomett. We furthermore provide a new isoperimetric inequality for the lattice point enumerator that generalizes previous results, and for which the aforementioned characterization also holds.
{"title":"On a Characterization of Lattice Cubes via Discrete Isoperimetric Inequalities","authors":"D. Iglesias, E. Lucas","doi":"10.37236/11024","DOIUrl":"https://doi.org/10.37236/11024","url":null,"abstract":"We obtain a characterization of lattice cubes as the only sets that reach equality in several discrete isoperimetric-type inequalities associated with the $L_{infty}$ norm, including well-known results by Radcliffe and Veomett. We furthermore provide a new isoperimetric inequality for the lattice point enumerator that generalizes previous results, and for which the aforementioned characterization also holds.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"2013 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78460606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It was proved by Scott that for every $kge 2$, there exists a constant $c(k)>0$ such that for every bipartite $n$-vertex graph $G$ without isolated vertices, there exists an induced subgraph $H$ of order at least $c(k)n$ such that $operatorname{deg}_H(v) equiv 1pmod{k}$ for each $v in H$. Scott conjectured that $c(k) = Omega(1/k)$, which would be tight up to the multiplicative constant. We confirm this conjecture.
Scott证明了对于每一个$kge 2$，存在一个常数$c(k)>0$，使得对于每一个没有孤立顶点的二部$n$顶点图$G$，存在一个至少为$c(k)n$阶的诱导子图$H$，使得$operatorname{deg}_H(v) equiv 1pmod{k}$对于每一个$v in H$。斯科特推测$c(k) = Omega(1/k)$，它会紧挨着乘法常数。我们证实了这个猜想。
{"title":"A Result on Large Induced Subgraphs with Prescribed Residues in Bipartite Graphs","authors":"Zachary Hunter","doi":"10.37236/11454","DOIUrl":"https://doi.org/10.37236/11454","url":null,"abstract":"It was proved by Scott that for every $kge 2$, there exists a constant $c(k)>0$ such that for every bipartite $n$-vertex graph $G$ without isolated vertices, there exists an induced subgraph $H$ of order at least $c(k)n$ such that $operatorname{deg}_H(v) equiv 1pmod{k}$ for each $v in H$. Scott conjectured that $c(k) = Omega(1/k)$, which would be tight up to the multiplicative constant. We confirm this conjecture.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"36 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72475762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Radoslav Fulek, Michael Pelsmajer, Marcus Schaefer
A drawing of a graph $G$, possibly with multiple edges but without loops, is radial if all edges are drawn radially, that is, each edge intersects every circle centered at the origin at most once. $G$ is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of $G$ are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the distances of the vertices from the origin respect the ordering or leveling.
A pair of edges $e$ and $f$ in a graph is independent if $e$ and $f$ do not share a vertex. We show that if a leveled graph $G$ has a radial drawing in which every two independent edges cross an even number of times, then $G$ is radial planar. In other words, we establish the strong Hanani-Tutte theorem for radial planarity. This characterization yields a very simple algorithm for radial planarity testing.
{"title":"Hanani-Tutte for Radial Planarity II","authors":"Radoslav Fulek, Michael Pelsmajer, Marcus Schaefer","doi":"10.37236/10169","DOIUrl":"https://doi.org/10.37236/10169","url":null,"abstract":"A drawing of a graph $G$, possibly with multiple edges but without loops, is radial if all edges are drawn radially, that is, each edge intersects every circle centered at the origin at most once. $G$ is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of $G$ are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the distances of the vertices from the origin respect the ordering or leveling.
 A pair of edges $e$ and $f$ in a graph is independent if $e$ and $f$ do not share a vertex. We show that if a leveled graph $G$ has a radial drawing in which every two independent edges cross an even number of times, then $G$ is radial planar. In other words, we establish the strong Hanani-Tutte theorem for radial planarity. This characterization yields a very simple algorithm for radial planarity testing.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"259 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135793917","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $V_1, V_2, V_3, dots $ be a sequence of $mathbb {Q}$-vector spaces where $V_n$ carries an action of $mathfrak{S}_n$. Representation stability and multiplicity stability are two related notions of when the sequence $V_n$ has a limit. An important source of stability phenomena arises when $V_n$ is the $d^{th}$ homology group (for fixed $d$) of the configuration space of $n$ distinct points in some fixed topological space $X$. We replace these configuration spaces with moduli spaces of tuples $(W_1, dots, W_n)$ of subspaces of a fixed complex vector space $mathbb {C}^N$ such that $W_1 + cdots + W_n = mathbb {C}^N$. These include the varieties of spanning line configurations which are tied to the Delta Conjecture of symmetric function theory.
{"title":"Spanning Configurations and Representation Stability","authors":"Brendan Pawlowski, Eric Ramos, B. Rhoades","doi":"10.37236/11136","DOIUrl":"https://doi.org/10.37236/11136","url":null,"abstract":"Let $V_1, V_2, V_3, dots $ be a sequence of $mathbb {Q}$-vector spaces where $V_n$ carries an action of $mathfrak{S}_n$. Representation stability and multiplicity stability are two related notions of when the sequence $V_n$ has a limit. An important source of stability phenomena arises when $V_n$ is the $d^{th}$ homology group (for fixed $d$) of the configuration space of $n$ distinct points in some fixed topological space $X$. We replace these configuration spaces with moduli spaces of tuples $(W_1, dots, W_n)$ of subspaces of a fixed complex vector space $mathbb {C}^N$ such that $W_1 + cdots + W_n = mathbb {C}^N$. These include the varieties of spanning line configurations which are tied to the Delta Conjecture of symmetric function theory.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"26 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83311638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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